I was just curious whether batch size has any bearing on the time requirements for any part of the brewing process. For example, does a one-gallon batch ferment any more quickly than a ten-gallon batch, all else being equal? Same question regarding aspects such as dry hopping, etc.

No the times are all pretty much the same no matter the batch size. A set number of yeast to eat a set amount of sugars will take the same amount of time no matter a gallon size or ten gallon size since the ratio of yeast to sugar does not change.

To answer your direct question, fermentation will take the same amount of time, regardless of batch size, assuming you pitch a proportionally appropriate amount of yeast. If you tried to ferment a 15 gallon batch with the same amount of yeast that you would use on a 5 gallon batch, you will notice that it takes longer (also there may be other problems). But if you pitched that same amount of yeast into a 1 gallon batch, you would likely blow the airlock out of the top.

Dry hopping results will occur just as quickly, but again, it will require a quantity of hops proportional to the batch size.

The major difference is in heat application. The larger the batch size, the longer it will take to heat water and cool wort.

After thinking about it though I seem to remember Revvy saying that a large bottle of beer carbs slower than a small bottle of beer. I have no knowledge of this in practical terms since I bottle 100% in 12 ounce bottles.

The bottle thing is going to factor more on the ratio of head-space to beer volume. The quantity of active yeast cells to the quantity of priming sugar should be the same, so the ratio of CO2 produced to the quantity of beer will be the same.

The difference would be the amount head-space left in the bottle. Obviously a 12 oz bottle will have a greater amount of head space than a bomber or (let's just say) a growler, when compared to the volume of beer it contains (22 and 64 oz). Any air space left above the beer must be pressurized to equilibrium before the CO2 would stay in solution. Based upon this, I would assume that a larger container would naturally carbonate faster.

But this is just speculation on my part as I am trying to perform this thought exercise with a girlfriend, a dog and two kids all simultaneously attempting to make as much noise as possible.

Oh, but on the subject of quantity vs. time, another thought occurs to me. The more beer you brew, the longer it generally takes to consume; cheers!